On reflection symmetry and its application to the Euler-Lagrange equations in fractional mechanics.

We study the properties of fractional differentiation with respect to the reflection symmetry in a finite interval. The representation and integration formulae are derived for symmetric and anti-symmetric fractional derivatives, both of the Riemann-Liouville and Caputo type. The action dependent on the left-sided Caputo derivatives of orders in the range (1,2) is considered and we derive the Euler-Lagrange equations for the symmetric and anti-symmetric part of the trajectory. The procedure is illustrated with an example of the action dependent linearly on fractional velocities. For the obtained Euler-Lagrange system, we discuss its localization resulting from the subsequent symmetrization of the action.